Remark 2.4 (Smoothing to eliminate asymptotic bias) 
Note the right hand side of Step (REF ) is not normalized by \(\frac{1}{\sqrt{n}}\) . In other words, with just a scaling of \(\sqrt{n}\)  on \(\sup _{h \in \mathcal {J}_n} \frac{\langle \hat{\rho } - \hat{\Pi }_{k_n}\rho ,h \rangle }{\sigma _\varepsilon t_n(h)}\) , its asymptotic bias terms do not converge to zero. In contrast to [1]}, here we do not benefit from the extra smoothing in a prediction problem \(\langle \hat{\rho } - \hat{\Pi }_{k_n}\rho ,X_{n+1} \rangle \) , where they show normalizing by just \(\sqrt{n}\)  is sufficient to ensure the asymptotic bias terms will vanish; see also [2]}.
